Application of adjoint-based optimization in three-dimensional flows interacting with multiple moving bodies

Abstract

Throughout nature, bodies in motion rarely move independent of each other but are often not studied in this multiple body form. This is often due to the complex motion exhibited by these bodies such as flocks of birds, pods of whales, or schools of fish. However, on a more fundamental level, the control space required to study any of these multiple body cases is immense and computationally unfeasible using traditional approaches. The continuous adjoint-based approach applied to a three-dimensional, multiple body case allows for a computationally feasible approach to study the complex flow-structure interactions for optimization and physical understanding. The computational cost associated with an adjoint-based approach is independent of the number of control parameters, making it an ideal method to solve complex problems with large control space. The traditional formulation of the adjoint equations utilize a fluid domain with fixed solid boundaries. However, with the introduction of moving solid boundaries, inconsistencies and ambiguity arise from the interaction between the Eulerian fluid domain and Lagrangian solid boundary when perturbation is considered at the solid boundary. Traditional methods utilize an unstable mapping function to mitigate the challenges of a moving boundary but increase the computational cost drastically and tend to be too complex to feasibly derive. To bypass the complexity required to use an unstable mapping function, application of non-cylindrical calculus allows for the simplification of the mapped domain to only the moving solid boundary. This approach, validated previously by the research group, is applied to study the broadening of the approach to encompass multiple bodies in a three-dimensional fluid flow and inherent challenges in implementation. The adjoint-based approach is first applied to optimize the motion of a pair of spheroids in an echelon formation to identify lift-generating regions behind a heaving leading body. The echelon pair case also explores the effect of variations in size of the trailing body. The lift-generation results identify a region directly behind a diving leading body and external vortex wall that can attribute over a relative 500% increase in force. Future work applies the approach to successively and collectively optimize a formation of spheroids and identify challenges in application to optimization of several bodies simultaneously.